Hilbert space representation

Hilbert space representations are an important mathematical method for the investigation of Banach - * - algebras , in particular C * -algebras and convolutionalgebras of locally compact groups . These are representations as algebras of operators on Hilbert spaces .

Compared to the general representation theory considered in algebra , there are additional structural elements due to the Hilbert space structure. First there is the topology of the Hilbert space, which also creates a topology in the space of continuous linear operators. The involution in the algebra of continuous linear operators on a Hilbert space, which is given by the adjunction , plays another important role .

Definitions

If a Hilbert space, then the algebra of continuous linear operators with the operator norm is a C * -algebra. ${\displaystyle H}$ ${\displaystyle L(H)}$

If a Banach - * - algebra , then every * - homomorphism is called a representation of on . ${\displaystyle A}$ ${\displaystyle \pi \colon A\rightarrow L(H)}$ ${\displaystyle A}$ ${\displaystyle H}$

With this definition, every representation is already a contraction with respect to the norm of the Banach - * - algebra and the operator norm on and thus continuous . ${\displaystyle L(H)}$

Many concept formations known from algebraic representation theory have topological variants, which are presented in the next section.

Further terminology

A Hilbert space representation is called cyclic if there is a vector such that it is close . ${\displaystyle \pi }$ ${\displaystyle \xi \in H}$ ${\displaystyle \{\pi (a)\xi ;\,a\in A\}\subset H}$

A subspace is called invariant (with regard to the representation ), if for all . If it is closed, i.e. a Hilbert space itself, then it is again a Hilbert space representation, it is called the corresponding partial representation . Since we are dealing here with * -homomorphisms, the orthogonal complement is also invariant. Hence, each is the direct sum of the operators and . For this one writes briefly , one then speaks of a direct sum of partial representations. Closed invariant subspaces allow the representation to be broken down into representations on smaller spaces. ${\displaystyle U\subset H}$ ${\displaystyle \pi }$ ${\displaystyle \pi (a)U\subset U}$ ${\displaystyle a\in A}$ ${\displaystyle U}$ ${\displaystyle \pi |_{U}\colon A\rightarrow L(U),\,\pi |_{U}(a):=\pi (a)|_{U}\colon U\rightarrow U}$ ${\displaystyle U}$ ${\displaystyle U^{\perp }}$ ${\displaystyle \pi (a)}$ ${\displaystyle \pi (a)|_{U}\,}$ ${\displaystyle \pi (a)|_{U^{\perp }}}$ ${\displaystyle \pi =\pi |_{U}\oplus \pi |_{U^{\perp }}}$

The smallest "building blocks" of Hilbert space representations are those that have no closed invariant subspaces except and . Such representations are called topologically irreducible . ${\displaystyle H}$ ${\displaystyle \{0\}}$ ${\displaystyle H}$

The null representation is the null homomorphism . A representation is called non-degenerate or non-degenerate if there is no closed invariant subspace, so that the restriction to this is the null representation. Each representation is the direct sum of cyclical partial representations and a zero representation. A non-degenerate representation is therefore a direct sum of cyclic representations and vice versa. ${\displaystyle A\rightarrow L(H)}$ ${\displaystyle \{0\}}$

Two representations and are called equivalent if there is a unitary operator such that for all . There is practically no difference between equivalent representations, only the names for the Hilbert space vectors (assets ) have been exchanged. ${\displaystyle \pi _{1}\colon A\rightarrow L(H_{1})}$ ${\displaystyle \pi _{2}\colon A\rightarrow L(H_{2})}$ ${\displaystyle U\colon H_{1}\rightarrow H_{2}}$ ${\displaystyle \pi _{1}(a)\,=\,U^{*}\pi _{2}(a)U}$ ${\displaystyle a\in A}$ ${\displaystyle U}$

There are very many representations, at least one for every cardinal number , namely the zero representation on a Hilbert space with a basis of this cardinality, and all of these representations are pairwise not equivalent. So one cannot speak of the set of equivalence classes of representations. This is different with irreducible representations that are small in a certain sense. If a C * -algebra or a group algebra, then the equivalence classes of irreducible representations form a set that is written and called the spectrum of . ${\displaystyle A}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle A}$

The kernels of irreducible representations are ideals that are called primitive . It is clear that equivalent representations lead to the same primitive ideal, the converse does not apply, but it does for postliminal C * -algebras . The space of primitive ideals is designated with. You then have a surjective mapping . Furthermore, primitive ideals are prime ideals . Hence the space of primitive ideals bears the relative Zariski topology . The initial topology with regard to the mapping is then the topology usually considered on the spectrum of . ${\displaystyle {\mbox{Prim}}(A)}$ ${\displaystyle {\hat {A}}\rightarrow {\mbox{Prim}}(A),\,[\pi ]\mapsto {\mbox{ker}}(\pi )}$ ${\displaystyle {\hat {A}}\rightarrow {\mbox{Prim}}(A)}$ ${\displaystyle A}$

Sentences about representations

GNS construction

→ Main article : State (mathematics)

A state on a Banach - * - algebra with an approximation of one bounded by 1 is a continuous linear functional with and for all . To such a state can be constructed as follows an illustration: To state set . Then the formula defines a scalar product on the quotient space . The completion with regard to this scalar product is a Hilbert space . For each imaging can be a continuous linear operator on continue. Then one shows that the mapping explained in this way is a cyclic representation with respect to the strong operator topology . This construction from is called the GNS construction according to Gelfand , Neumark and Segal (see also the Gelfand-Neumark theorem ), and is also called the GNS representation of the state . ${\displaystyle A}$ ${\displaystyle (e_{i})_{i}}$ ${\displaystyle f\colon A\rightarrow \mathbb {C} }$ ${\displaystyle \|f\|=1}$ ${\displaystyle f(a^{*}a)\geq 0}$ ${\displaystyle a\in A}$ ${\displaystyle f}$ ${\displaystyle N_{f}:=\{x\in A:f(x^{*}x)=0\}}$ ${\displaystyle \langle x+N_{f},y+N_{f}\rangle =f(y^{*}x)}$ ${\displaystyle A/N_{f}}$ ${\displaystyle H_{f}}$ ${\displaystyle a\in A}$ ${\displaystyle x+N_{f}\mapsto ax+N_{f}}$ ${\displaystyle \pi _{f}(a)}$ ${\displaystyle H_{f}}$ ${\displaystyle \pi _{f}\colon A\rightarrow L(H_{f})}$ ${\displaystyle \pi _{f}(e_{i})\rightarrow {\mbox{id}}_{H_{f}}}$ ${\displaystyle \pi _{f}}$ ${\displaystyle f}$ ${\displaystyle \pi _{f}}$ ${\displaystyle f}$

Enveloping C * algebra, enveloping Von Neumann algebra

Let be a Banach - * - algebra with an approximation of one bounded by 1. The direct sum of all GNS representations , where the set of all states passes through, is called the universal representation of . ${\displaystyle A}$ ${\displaystyle \pi _{f}}$ ${\displaystyle f}$ ${\displaystyle \pi _{u}}$ ${\displaystyle A}$

${\displaystyle \pi _{u}}$is a non-degenerate representation on the Hilbert space . In the case of C * algebras and group algebras, the universal representation is faithful ( i.e., injective ). The closure of with regard to the standard topology is called the enveloping C * -algebra of . The closure of with respect to the weak operator topology contains the operator and is called the enveloping Von-Neumann algebra of . The enveloping Von Neumann algebra of a C * algebra can be identified with its bidual, provided with the Arens product . ${\displaystyle H_{u}:=\bigoplus H_{f}}$ ${\displaystyle \pi _{u}(A)\subset L(H_{u})}$ ${\displaystyle A}$ ${\displaystyle \pi _{u}(A)\subset L(H_{u})}$ ${\displaystyle {\mbox{id}}_{H_{u}}}$ ${\displaystyle A}$

Existence of irreducible representations, atomic representation

It is not clear a priori whether there are any irreducible representations in the case of C * -algebras or group algebras . In fact, it is so that at any one irreducible representation are with how to prove by means of the GNS construction. It follows immediately that the direct sum is a true representation. This special representation, which contains exactly one equivalent partial representation for every irreducible representation, is called the atomic representation . ${\displaystyle A}$ ${\displaystyle 0\not =a\in A}$ ${\displaystyle \pi }$ ${\displaystyle \pi (a)\not =0}$ ${\displaystyle \textstyle \bigoplus _{[\pi ]\in {\hat {A}}}\pi }$

Some classes of C * algebras are characterized by their irreducible representations:

• A C * -algebra is commutative if and only if every irreducible representation is one-dimensional.
• A C * -algebra is called CCR-algebra ( completely continuous representations ) if the image of each irreducible representation is the same as the algebra of the compact operators .
• A C * -algebra is called GCR-algebra ( generalized completely continuous representations ) if the image of each irreducible representation includes the algebra of the compact operators.

Kadison's transitivity theorem

A representation is called algebraically irreducible if, apart from and, there are no invariant subspaces, i.e. also no non-closed subspaces. Algebraic irreducibility is therefore the stronger requirement; however, Kadison's transitivity theorem applies , which can be proven with the help of Kaplansky's theorem of density : ${\displaystyle \{0\}}$ ${\displaystyle H}$

• Let be a C * -algebra and a topologically irreducible representation. Further let be linearly independent and . Then there is one with for everyone . ${\displaystyle A}$ ${\displaystyle \pi \colon A\rightarrow L(H)}$ ${\displaystyle \xi _{1},\ldots ,\xi _{n}\in H}$ ${\displaystyle \eta _{1},\ldots ,\eta _{n}\in H}$ ${\displaystyle a\in A}$ ${\displaystyle \pi (a)\xi _{i}\,=\,\eta _{i}}$ ${\displaystyle i=1,\ldots ,n}$

This immediately results in the following corollary:

• A representation of a C * -algebra is topologically irreducible if and only if it is algebraically irreducible.

It only needs to be shown that topologically irreducible representations are also algebraically irreducible, because the converse is clear. Is one of an invariant of different vector space, so there is a different from 0 . Is arbitrary, there is a with after the transitivity theorem . Since is invariant, it follows , so in total . The only invariant subspaces are therefore and , that is, there is algebraic irreducibility. ${\displaystyle U\subset H}$ ${\displaystyle \{0\}}$ ${\displaystyle \xi \in U}$ ${\displaystyle \eta \in H}$ ${\displaystyle a\in A}$ ${\displaystyle \pi (a)\xi =\eta }$ ${\displaystyle U}$ ${\displaystyle \eta \in U}$ ${\displaystyle U=H}$ ${\displaystyle \{0\}}$ ${\displaystyle H}$

literature

• W. Arveson : Invitation to C * -algebras , ISBN 0-387-90176-0
• J. Dixmier : Les C * -algèbres et leurs représentations , Gauthier-Villars, 1969
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , 1983, ISBN 0-123-93301-3

Individual evidence

1. ^ William Arveson : A Short Course on Spectral Theory . Springer , New York 2002, ISBN 0-387-95300-0 , pp. 58 , doi : 10.1007 / b97227 .